Abstract and Applied Analysis
Volume 2016 (2016), Article ID 3094838, 6 pages
http://dx.doi.org/10.1155/2016/3094838

Research Article

Approximation of a Common Element of the Fixed Point Sets of Multivalued Strictly Pseudocontractive-Type Mappings and the Set of Solutions of an Equilibrium Problem in Hilbert Spaces

F. O. Isiogugu1,2

¹School of Mathematics, Statistics and Computer Sciences, University of KwaZulu-Natal, Westville Campus, Durban 4000, South Africa
²Department of Mathematics, University of Nigeria, Nsukka 410001, Nigeria

Received 8 December 2015; Revised 14 February 2016; Accepted 17 February 2016

Academic Editor: Simeon Reich

Copyright © 2016 F. O. Isiogugu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The strong convergence of a hybrid algorithm to a common element of the fixed point sets of multivalued strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces is obtained using a strict fixed point set condition. The obtained results improve, complement, and extend the results on multivalued and single-valued mappings in the contemporary literature.

1. Introduction

Let be a nonempty set and let be a map. A point is called a fixed point of if . If is a multivalued map then is a fixed point of if . If then is called a strict fixed point of . The set (resp., ) is called the fixed point set of multivalued (resp., single-valued) map , while the set is called the strict fixed point set of .

Let be a normed space. A subset of is called proximinal if for each there exists such that

It is known that every closed convex subset of a uniformly convex Banach space is proximinal. We will denote the family of all nonempty closed and bounded subsets of by , the family of all nonempty subsets of by , and the family of all proximinal subsets of by , for a nonempty set .

Let denote the Hausdorff metric induced by the metric on ; that is, for every ,

Let be a normed space. Let be a multivalued mapping on . A multivalued mapping is called -Lipschitzian if there exists such that for all

In (3) if is said to be a contraction while is nonexpansive if . is called quasi-nonexpansive if and for all ,

Clearly every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive. is said to be -strictly pseudocontractive-type of Isiogugu [1] if there exists such that, given any pair and , there exists satisfying and

If in (5), is said to be pseudocontractive-type, while is nonexpansive-type if . Every multivalued nonexpansive mapping is nonexpansive-type. In a real Hilbert space , is said to be -strictly pseudocontractive of Chidume et al. [2] if there exists such that for all

If , is said to be pseudocontractive. It is easy to see that every -strictly pseudocontractive mapping is -strictly pseudocontractive-type.

Let be a real Hilbert space with inner product and norm , respectively, and let be a nonempty closed convex subset of . Given an operator and a closed convex set , the variational inequality problem is the problem of finding such that , for all . This variational inequality problem is usually denoted as .

Let be a bifunction, where is the set of real numbers. The equilibrium problem for is to find such that

The set of solutions of (7) is denoted by . Several algorithms were introduced by authors for approximating solutions of equilibrium problems for a bifunction (or finite family of bifunctions) (see, e.g., [3] and references therein). Given a mapping , let for all ; then if and only if for all ; that is, is a solution of the variational inequality . Numerous problems in physics, optimization, and economics are reduced to the problem of finding the solutions of (7) (see, e.g., [4–6] and the references therein).

The purpose of this work is to first establish closed and convexity property for a strict fixed point set of a multivalued strictly pseudocontractive-type mappings. Second, establish with a strict fixed point set condition a strong convergence of a hybrid algorithm to a common element of the fixed point sets of two multivalued strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces. The obtained results extend, complement, and improve the results on equilibrium problems as well as multivalued and single-valued mappings in the contemporary literature.

2. Preliminaries

In the sequel, we will need the following definitions and lemmas.

Definition 1. Let be a multivalued mapping; for each , is defined by

For solving the equilibrium problems for a bifunction , let us assume that satisfies the following conditions:

(A1) for all .(A2) is monotone; that is, , for all (A3)For each , .(A4)For each , is convex and lower semicontinuous.

Lemma 2 (see [4]). Let be a nonempty closed convex subset of a real Hilbert space and a bifunction satisfying (A1)–(A4). Let and . Then, there exists such that

Lemma 3 (see [6]). Let be a nonempty closed convex subset of a real Hilbert space . Assume that satisfying (A1)–(A4). Let and . Define by

Then the following hold:

(1) is single valued.(2) is firmly nonexpansive; that is, for any , .(3).(4) is closed and convex.

Lemma 4 (see [7]). Let be a nonempty closed convex subset of a real Hilbert space and a bifunction satisfying (A1)–(A4). Let and . Then for all and

Lemma 5. Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let be the convex projection onto . Then, convex projection is characterized by the following relations:

(i), for all (ii).(iii).

3. Main Results

Proposition 6. Let be a nonempty subset of a real Hilbert space . And let be a -strictly pseudocontractive-type mapping such that is nonempty. Then is closed and convex.

Proof. Let such that converges to . We show that . Let be arbitrary:

Taking limits as , we have that . Hence, . Since was arbitrary, we have that .
We now prove that is convex. Let and and then and :

Now, -strictly pseudocontractive-type condition on and a strict fixed point condition on and imply that, for all , and and and . It then follows that

In particular, for each ,

Hence, . Since is proximinal, there exists such that ; consequently, . Also, if , then

which shows that . Thus, .

We now prove a strong convergence of multivalued version of the hybrid algorithm considered in [8] to a common element of the set of fixed points of two -strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces. As a corollary, we obtain a hybrid algorithm for finding common elements of the set of fixed points of two multivalued strictly pseudocontractive mappings of [2] and the set of solutions of an equilibrium problem, with a strict fixed point set condition.

Theorem 7. Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A4), and let be two strictly pseudocontractive-type mappings with contractive coefficients and , respectively, such that . Let be a sequence generated from an arbitrary as follows:

where and . and are sequences in satisfying

(i),(ii) and ,(iii) for some .

Then converges strongly to .

Proof. Observe that is closed and convex for all ; therefore is well defined and note that . Next we show that , for all . is obvious. Suppose , set , and then using Lemma 3, for all , we have

Also,

Using (19) we obtain from (18) that

Also,

Using (21) we obtain from (20) that

This shows that . It then follows that for all . From we have from Lemma 5(i) that

Since for all , we have

Using Lemma 5(ii) we obtain

for each and for all . Consequently the sequence is bounded, and so are and . Furthermore, since and then from definition of we have for all . Therefore the sequence is nondecreasing. It then follows that exists. From the construction of we have that and for any integer . It also follows from Lemma 5(iii) that

Letting in (26), we have . Hence is a Cauchy sequence. Since is Hilbert and is closed and convex we can assume that as ; that is, . We now show that . In particular when in (26) we obtain

Also, since , we obtain

It then follows from (27) that

Combining (27) and (29) we obtain

It follows from and (30) that

Setting in (22) we have

Observe that

It then follows from (30) that

Using and we obtain from (32) that and . Hence . It remains to show that is in . Now from (32)

Also, using , Lemma 4, and (35) we have

It then follows from (34) and (36) that

Consequently, we obtain from (31) and (37) that

From the assumption that ,

Since implies

we have from (A2) that

By taking limit as of the above inequality and from (A4), (31), and (38) we have , for all . Let and for all , since , we have that . Hence . It follows from (A1) that

that is, . Letting , from (A3) we obtain for all so that . Hence .
Finally we show that . By taking the limits as in (23) we have

It then follows from Lemma 5(i) that . This completes the proof.

Corollary 8. Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A4), and let be two strictly pseudocontractive mappings with contractive coefficients and , respectively, such that . Let be a sequence generated from an arbitrary as follows:

where and . and are sequences in satisfying

(i),(ii) and ,(iii) for some .

Then converges strongly to .

Proof. The proof follows easily from Theorem 7.

Competing Interests

The author declares that there are no competing interests.

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